Question

Solution of the differential equation \( \frac{dy}{dx} + x = 0 \) represents a family of

• A. Ellipse
• B. Parabola
• C. Circles
• D. Hyperbola

Hint:

When solving first-order linear differential equations, look for the general solution which typically involves an arbitrary constant. The form of the solution helps identify the family of curves it represents.

Answer 1

The Correct Option is C

Step 1: Analyze the given differential equation.
We are given the first-order linear differential equation: \[ \frac{dy}{dx} + x = 0 \]
This is a simple linear equation where \( \frac{dy}{dx} \) represents the rate of change of \( y \) with respect to \( x \), and the equation relates the two variables.

Step 2: Rearrange the equation.
Rearranging the equation, we get:
\[ \frac{dy}{dx} = -x \] This implies that the rate of change of \( y \) is a linear function of \( x \), and the slope of the function is negative.

Step 3: Integrate both sides.
Now, we integrate both sides with respect to \( x \). The left-hand side is the derivative of \( y \) with respect to \( x \), so the integral gives:
\[ y = \int (-x) \, dx \]
The integral of \( -x \) is \( -\frac{x^2}{2} \), so we have:
\[ y = -\frac{x^2}{2} + C \]
where \( C \) is the constant of integration.

Step 4: Recognize the family of curves.
The equation \( y = -\frac{x^2}{2} + C \) represents a family of parabolas, where \( C \) is a parameter that determines the specific curve within the family. These curves open downward (as indicated by the negative sign in front of \( x^2 \)).

Step 5: Verify the solution.
This is indeed the general solution for a family of parabolas. The constant \( C \) determines the vertical shift of each parabola, meaning different values of \( C \) give different curves, all of which are parabolas.

Step 6: Conclusion.
Thus, the solution of the given differential equation represents a family of parabolas, confirming that the correct answer is option (B).
Final Answer: \[ \boxed{\text{Circles}} \]